Hexadecimal Calculator
Advanced Tool for Hex Arithmetic, Conversions, and Bitwise Operations
Quick Navigation
Hexadecimal Arithmetic
Add, subtract, multiply, or divide hexadecimal numbers
Number System Conversion
Convert between hexadecimal, decimal, binary, and octal
Bitwise Operations on Hex
Perform logical operations on hexadecimal numbers
Multi-Format Display
View the same number in all base formats
What is Hexadecimal?
Hexadecimal (hex) is a base-16 number system using 16 digits: 0-9 and A-F (where A=10, B=11, C=12, D=13, E=14, F=15). It's one of the most important number systems in computing because it provides a compact, human-readable way to represent binary data.
While computers work internally with binary (base 2), hexadecimal serves as a bridge between human understanding and machine representation. One hexadecimal digit represents exactly 4 binary digits (a nibble), making conversion trivial. One byte (8 bits) equals two hex digits, which is why hex dominates in programming and system administration.
Hexadecimal appears everywhere in computing: memory addresses, color codes in web design (#FF00FF), byte values, MAC addresses, IP subnets, Unicode characters, and machine instructions. Understanding hexadecimal is essential for any technical work with computers.
Key Concept: Hexadecimal is just another way to represent numbers. The number remains the same regardless of base: 255₁₀ = FF₁₆ = 11111111₂. They all represent the same quantity, just written differently.
Key Features & Capabilities
This comprehensive hexadecimal calculator provides multiple calculation modes and detailed analysis:
➕ Hex Arithmetic
Add, subtract, multiply, and divide hexadecimal numbers with step-by-step solutions
🔄 Number Conversions
Convert between hexadecimal, decimal, binary, and octal with detailed process breakdown
⚙️ Bitwise Operations
Perform AND, OR, XOR operations on hexadecimal numbers for low-level programming
📊 Multi-Format Display
View same number in hexadecimal, decimal, binary, and octal simultaneously
📋 Step-by-Step Breakdown
See detailed calculation steps showing exactly how results are obtained
✓ Input Flexibility
Accept hex (0-9, A-F), decimal, and binary input with automatic format detection
🔢 Case Insensitivity
Accept both uppercase (FF) and lowercase (ff) hex notation automatically
📋 Copy to Clipboard
One-click copy functionality to transfer results to other applications
🎓 Educational Content
Comprehensive guides, examples, and explanations of hexadecimal concepts
⚡ Real-Time Calculation
Instant results with no delays or external dependencies
📱 Fully Responsive
Works seamlessly on desktop, tablet, and mobile devices
🔧 Professional Features
Support for large hex numbers, negative values, and complementary operations
How to Use This Calculator
Step-by-Step Guide
- Select Your Operation: Click the appropriate tab: Arithmetic for hex math, Convert for base conversions, Bitwise for logical operations, or Format for multi-format display.
- Enter Your Values: Input hexadecimal numbers using 0-9 and A-F (case insensitive). The calculator accepts both uppercase and lowercase notation.
- Configure Parameters: Select the specific operation: addition, subtraction, multiplication, division for arithmetic; target format for conversions; AND, OR, XOR for bitwise.
- Click Calculate: Press the Calculate button to perform the computation using proper hexadecimal algorithms.
- Review Results: The main result displays prominently in hexadecimal (and often in other formats too).
- Study Steps: Below the result, see detailed step-by-step breakdown showing the calculation process.
- View Statistics: See results displayed in multiple formats (hex, decimal, binary, octal) for comprehensive understanding.
- Copy or Clear: Use Copy to transfer results. Use Clear to reset for a new calculation.
Tips for Accurate Use
- Hex Notation: Hexadecimal uses digits 0-9 and letters A-F. The letters can be uppercase or lowercase (both work equally).
- 0x Prefix: Some notations use 0x prefix (like 0xFF). This calculator accepts both with and without the prefix.
- Overflow Handling: Arithmetic operations might produce results larger than typical byte size. This is normal and expected.
- Leading Zeros: Hex numbers can have leading zeros without affecting value: 00FF = 0FF = FF = FF₁₆ all represent 255 in decimal.
- Negative Numbers: Use minus sign for negatives. Two's complement representation is available when needed.
Complete Formulas Guide
Hexadecimal Number System
Positional Value in Hexadecimal
Value = (d_n × 16^n) + (d_(n-1) × 16^(n-1)) + ... + (d_1 × 16^1) + (d_0 × 16^0)Where d represents each digit (0-F) and position increases from right to left.
Example: 1A3₁₆ = (1×16²) + (10×16¹) + (3×16⁰) = 256 + 160 + 3 = 419₁₀
Hex to Decimal Conversion
Converting Hexadecimal to Decimal
Multiply each digit by its place value (power of 16) and sum.
Example: FF₁₆ = (F×16¹) + (F×16⁰) = (15×16) + (15×1) = 240 + 15 = 255₁₀
Letter Conversions:
A=10, B=11, C=12, D=13, E=14, F=15
Example: FF₁₆ = (F×16¹) + (F×16⁰) = (15×16) + (15×1) = 240 + 15 = 255₁₀
Letter Conversions:
A=10, B=11, C=12, D=13, E=14, F=15
Decimal to Hex Conversion
Converting Decimal to Hexadecimal
Repeatedly divide by 16, record remainders, convert remainders to hex (10-15 become A-F).
Example: 255₁₀ to Hex
255 ÷ 16 = 15 remainder 15 (F)
15 ÷ 16 = 0 remainder 15 (F)
Read remainder bottom to top: FF₁₆
Example: 255₁₀ to Hex
255 ÷ 16 = 15 remainder 15 (F)
15 ÷ 16 = 0 remainder 15 (F)
Read remainder bottom to top: FF₁₆
Hex Addition Rules
Hexadecimal Addition Process
Add digit by digit from right to left.
If sum ≤ 15 (F): write the hex digit
If sum > 15: subtract 16 and carry 1 to next column
Example: A5₁₆ + 3F₁₆
5 + F = 5 + 15 = 20 = 16 + 4, write 4 carry 1
A + 3 + 1(carry) = 10 + 3 + 1 = 14 = E
Result: E4₁₆
If sum ≤ 15 (F): write the hex digit
If sum > 15: subtract 16 and carry 1 to next column
Example: A5₁₆ + 3F₁₆
5 + F = 5 + 15 = 20 = 16 + 4, write 4 carry 1
A + 3 + 1(carry) = 10 + 3 + 1 = 14 = E
Result: E4₁₆
Hex and Binary Relationship
Hexadecimal to Binary Conversion
Each hex digit represents exactly 4 binary digits:
0=0000, 1=0001, 2=0010, 3=0011,
4=0100, 5=0101, 6=0110, 7=0111,
8=1000, 9=1001, A=1010, B=1011,
C=1100, D=1101, E=1110, F=1111
Example: AB₁₆ = 1010 1011₂
0=0000, 1=0001, 2=0010, 3=0011,
4=0100, 5=0101, 6=0110, 7=0111,
8=1000, 9=1001, A=1010, B=1011,
C=1100, D=1101, E=1110, F=1111
Example: AB₁₆ = 1010 1011₂
Conversion Methods
Why Hexadecimal is Used
Hexadecimal bridges the gap between binary's precision and decimal's readability. Since one hex digit = 4 binary digits, and one byte = 8 bits = 2 hex digits, hex provides an elegant way to represent byte values. For example, a single byte ranging from 0-255 in decimal becomes a neat 00-FF in hex.
Common Hex Applications
- Web Colors: #RRGGBB format (e.g., #FF0000 for red)
- Memory Addresses: 0x8000000 style notation
- MAC Addresses: 00-1A-2B-3C-4D-5E format
- Unicode Characters: U+0041 for 'A'
- Byte Values: 0x00 to 0xFF represent 0-255
- Programming Constants: 0xDEADBEEF (famous test value)
Hex Digit to Decimal Quick Reference
| Hex | Decimal | Hex | Decimal | Hex | Decimal |
|---|---|---|---|---|---|
| 0 | 0 | 4 | 4 | 8 | 8 |
| 1 | 1 | 5 | 5 | 9 | 9 |
| 2 | 2 | 6 | 6 | A | 10 |
| 3 | 3 | 7 | 7 | F | 15 |
Worked Examples
Example 1: Hex Addition
Problem: Add 1A₁₆ + 2B₁₆
Solution:
1A
+2B
---
45
Process:
A + B = 10 + 11 = 21 = 0x15 (write 5, carry 1)
1 + 2 + 1(carry) = 4
Result: 45₁₆
Verification: 26₁₀ + 43₁₀ = 69₁₀ = 45₁₆ ✓
1A
+2B
---
45
Process:
A + B = 10 + 11 = 21 = 0x15 (write 5, carry 1)
1 + 2 + 1(carry) = 4
Result: 45₁₆
Verification: 26₁₀ + 43₁₀ = 69₁₀ = 45₁₆ ✓
Example 2: Decimal to Hexadecimal
Problem: Convert 173₁₀ to hexadecimal
Solution:
173 ÷ 16 = 10 remainder 13 (D)
10 ÷ 16 = 0 remainder 10 (A)
Read remainders from bottom to top: AD₁₆
Verification: (A×16¹) + (D×16⁰) = (10×16) + 13 = 160 + 13 = 173₁₀ ✓
173 ÷ 16 = 10 remainder 13 (D)
10 ÷ 16 = 0 remainder 10 (A)
Read remainders from bottom to top: AD₁₆
Verification: (A×16¹) + (D×16⁰) = (10×16) + 13 = 160 + 13 = 173₁₀ ✓
Example 3: Hex Color Code
Problem: What RGB values does #3F7A05 represent?
Solution:
Split hex code into pairs:
3F (Red) = (3×16) + 15 = 48 + 15 = 63₁₀
7A (Green) = (7×16) + 10 = 112 + 10 = 122₁₀
05 (Blue) = (0×16) + 5 = 5₁₀
RGB values: (63, 122, 5) - A greenish color
Split hex code into pairs:
3F (Red) = (3×16) + 15 = 48 + 15 = 63₁₀
7A (Green) = (7×16) + 10 = 112 + 10 = 122₁₀
05 (Blue) = (0×16) + 5 = 5₁₀
RGB values: (63, 122, 5) - A greenish color
Example 4: Hex Subtraction
Problem: Calculate FF₁₆ - A7₁₆
Solution:
FF
-A7
---
58
Process:
F - 7 = 15 - 7 = 8
F - A = 15 - 10 = 5
Result: 58₁₆
Verification: 255₁₀ - 167₁₀ = 88₁₀ = 58₁₆ ✓
FF
-A7
---
58
Process:
F - 7 = 15 - 7 = 8
F - A = 15 - 10 = 5
Result: 58₁₆
Verification: 255₁₀ - 167₁₀ = 88₁₀ = 58₁₆ ✓
Example 5: Hex Multiplication
Problem: Calculate 12₁₆ × F₁₆
Solution:
Convert to decimal first (easier):
12₁₆ = 18₁₀
F₁₆ = 15₁₀
18 × 15 = 270₁₀
Convert back to hex:
270 ÷ 16 = 16 remainder 14 (E)
16 ÷ 16 = 1 remainder 0
1 ÷ 16 = 0 remainder 1
Result: 10E₁₆
Verification: (1×256) + (0×16) + 14 = 270₁₀ ✓
Convert to decimal first (easier):
12₁₆ = 18₁₀
F₁₆ = 15₁₀
18 × 15 = 270₁₀
Convert back to hex:
270 ÷ 16 = 16 remainder 14 (E)
16 ÷ 16 = 1 remainder 0
1 ÷ 16 = 0 remainder 1
Result: 10E₁₆
Verification: (1×256) + (0×16) + 14 = 270₁₀ ✓
Frequently Asked Questions
❓ Why is hex used more than octal in modern computing?
Hexadecimal is superior to octal for modern computing because one hex digit represents exactly 4 binary digits (2 decimal won't divide evenly), and one byte equals 2 hex digits (perfect alignment). This makes hex ideal for byte-oriented systems. Octal was more popular with 8-bit and 12-bit systems.
❓ Can hexadecimal have decimal points?
Yes! Hexadecimal can represent fractional values with a hex point (analogous to decimal point). For example, 0.8₁₆ = (8×16^-1) = 8/16 = 0.5₁₀. However, this is rare in computing practice.
❓ What is the largest hex digit?
F is the largest single hex digit, representing 15 in decimal. In multi-digit hex numbers, you can have as many F's as needed (FFF, FFFF, etc.), each representing the maximum value for that position.
❓ How do I remember hex letter values?
A=10, B=11, C=12, D=13, E=14, F=15. A helpful pattern: the letters go in order (A through F) from 10 to 15. You can also think "A is After 9" (10), and count up from there.
❓ What's the difference between 0x and $ prefix for hex?
Different programming languages use different conventions: 0x (used in C, Java, Python), $ (used in assembly, some BASIC), h or H suffix (some assembly dialects). They all mean hexadecimal. This calculator accepts 0x prefix and also notation without any prefix.
❓ Can I use hex for floating-point numbers?
Yes! Hexadecimal floating-point is used in some programming languages (like Java and Python). It follows similar patterns to decimal floating-point but uses base 16. Example: 0x1.8p+3 in hex float notation.
❓ How many bits does two hex digits represent?
Two hexadecimal digits represent exactly 8 bits (one byte). This is why hex is so convenient: one byte (0-255 decimal) becomes one byte value (00-FF hex). This perfect alignment doesn't exist with octal.
❓ What is a nibble?
A nibble is 4 bits, and it's represented perfectly by one hexadecimal digit (0-F). This term emphasizes how hex maps to binary: each hex digit is one nibble. Two nibbles = one byte = two hex digits.
❓ Can hex represent negative numbers?
Yes, through several methods: prefix with minus sign (-FF), use two's complement notation, or sign-magnitude representation. The method depends on the system. Most modern systems use two's complement for consistency with binary representation.
❓ How do MAC addresses use hex?
MAC addresses are 48-bit identifiers written in hex format: XX-XX-XX-XX-XX-XX (six pairs of hex digits). Each pair represents one byte. Example: 00-1A-2B-3C-4D-5E. Hex is perfect here since one byte = two hex digits.
❓ When should I use this calculator?
Use it for: programming (debugging, variable values), web design (color codes), networking (MAC/IP addresses), security work (encryption keys), reverse engineering, embedded systems, digital electronics, studying computer science, or any technical field working with machine-level data.
Start Working with Hexadecimal
Whether you're a programmer debugging code, a designer working with color values, a network administrator managing systems, or a student learning computer science, this comprehensive hex calculator helps you master hexadecimal operations. Fast, accurate, and completely free.